In the month of March, the temperature at the South Pole varies over the day in a periodic way that can be modeled approximately by a trigonometric function. The highest temperature is about $-50^\circ C$, and it is reached around $2\text{ p.m.}$ The lowest temperature is about $-54^\circ C$ and it is reached half a day apart from the highest temperature, at $2\text{ a.m.}$ Find the formula of the trigonometric function that models the temperature $T$ in the South Pole in March $t$ hours after midnight. Define the function using radians. $ T(t) = $ What is the temperature at $5\text{ p.m.}$ ? Round your answer, if necessary, to two decimal places. $ $
Explanation: Let's start by writing a formula for the temperature $u$ hours after $2\text{ p.m.}$ Both sine and cosine can be used to model periodic contexts. We can decide which is better fitting by considering the $y$ -intercept. The sine function intercepts the $y$ -axis at its midline, and the cosine function intercepts the $y$ -axis at its peak. We know the temperature reaches its highest point at $u = 0$, so let's use a cosine function. The amplitude of the temperature function is $ \dfrac{-50 - (-54)}{2} = 2^\circ C$. The period is $24$ hours, since the temperature reaches its peak once every $24$ hours. The midline is the average of the highest and lowest values, or $ \dfrac{(-50) + (-54)}{2} = -52^\circ C$ Since the ordinary cosine function $f(u) = \cos u$ has period $2\pi$, midline $y = 0$, and amplitude $1$, we need to stretch it horizontally by a factor of ${\dfrac{24}{2\pi}}$, stretch it vertically by a factor of ${2}$, and move it down by ${52}$ units. $ T(u) = {2}\cos\left({\dfrac{2\pi}{24}}u\right) - {52}$ Since $2\text{ p.m.}$ is $14$ hours after midnight, $t$ hours after midnight is $t -14$ hours after $2\text{ p.m.}$ So $u = t - 14$ : $ T(t) = {2}\cos\left({\dfrac{2\pi}{24}}(t-14)\right) - {52}$ At $5\text{ p.m.}$, $17$ hours after midnight, the temperature is $\begin{aligned} T(17) &= 2\cos\left(\dfrac{2\pi}{24}(17 - 14)\right) - 52\\ &\approx -50.59^\circ C\end{aligned}$ A correct formula for $T(t)$ is: $ T(t) = 2\cos\left(\dfrac{2\pi}{24}(t-14)\right) - 52$ The temperature at $5\text{ p.m.}$ is: $ -50.59^\circ C$